The number of parking functions with center of a given length
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/25846 |
Resumo: | Let 1≤r≤n and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labeled tree on n+1 vertices, exactly r vertices are visited before backtracking. Let R be the set of trees with this property. We count the number of elements of R. For this purpose, we first consider a bijection, due to Perkinson, Yang and Yu, that maps R onto the set of parking function with center (defined by the authors in a previous article) of size r. A second bijection maps this set onto the set of parking functions with run r, a property that we introduce here. We then prove that the number of length n parking functions with a given run is the number of length n rook words (defined by Leven, Rhoades and Wilson) with the same run. This is done by counting related lattice paths in a ladder-shaped region. We finally count the number of length n rook words with run r, which is the answer to our initial question. |
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The number of parking functions with center of a given lengthParking functionsShi arrangementIsh arrangementLet 1≤r≤n and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labeled tree on n+1 vertices, exactly r vertices are visited before backtracking. Let R be the set of trees with this property. We count the number of elements of R. For this purpose, we first consider a bijection, due to Perkinson, Yang and Yu, that maps R onto the set of parking function with center (defined by the authors in a previous article) of size r. A second bijection maps this set onto the set of parking functions with run r, a property that we introduce here. We then prove that the number of length n parking functions with a given run is the number of length n rook words (defined by Leven, Rhoades and Wilson) with the same run. This is done by counting related lattice paths in a ladder-shaped region. We finally count the number of length n rook words with run r, which is the answer to our initial question.Elsevier2020-06-01T00:00:00Z2019-06-01T00:00:00Z2019-06info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/25846eng0196-885810.1016/j.aam.2019.02.004Duarte, RuiGuedes de Oliveira, Antónioinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-22T11:50:02Zoai:ria.ua.pt:10773/25846Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:58:59.066700Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
The number of parking functions with center of a given length |
title |
The number of parking functions with center of a given length |
spellingShingle |
The number of parking functions with center of a given length Duarte, Rui Parking functions Shi arrangement Ish arrangement |
title_short |
The number of parking functions with center of a given length |
title_full |
The number of parking functions with center of a given length |
title_fullStr |
The number of parking functions with center of a given length |
title_full_unstemmed |
The number of parking functions with center of a given length |
title_sort |
The number of parking functions with center of a given length |
author |
Duarte, Rui |
author_facet |
Duarte, Rui Guedes de Oliveira, António |
author_role |
author |
author2 |
Guedes de Oliveira, António |
author2_role |
author |
dc.contributor.author.fl_str_mv |
Duarte, Rui Guedes de Oliveira, António |
dc.subject.por.fl_str_mv |
Parking functions Shi arrangement Ish arrangement |
topic |
Parking functions Shi arrangement Ish arrangement |
description |
Let 1≤r≤n and suppose that, when the Depth-first Search Algorithm is applied to a given rooted labeled tree on n+1 vertices, exactly r vertices are visited before backtracking. Let R be the set of trees with this property. We count the number of elements of R. For this purpose, we first consider a bijection, due to Perkinson, Yang and Yu, that maps R onto the set of parking function with center (defined by the authors in a previous article) of size r. A second bijection maps this set onto the set of parking functions with run r, a property that we introduce here. We then prove that the number of length n parking functions with a given run is the number of length n rook words (defined by Leven, Rhoades and Wilson) with the same run. This is done by counting related lattice paths in a ladder-shaped region. We finally count the number of length n rook words with run r, which is the answer to our initial question. |
publishDate |
2019 |
dc.date.none.fl_str_mv |
2019-06-01T00:00:00Z 2019-06 2020-06-01T00:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10773/25846 |
url |
http://hdl.handle.net/10773/25846 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
0196-8858 10.1016/j.aam.2019.02.004 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Elsevier |
publisher.none.fl_str_mv |
Elsevier |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
repository.name.fl_str_mv |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
repository.mail.fl_str_mv |
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1799137643990089728 |